Iraqi forces backed by Iranian-allied militias began an assault Thursday to reclaim more Kurdish-held territory in Iraq, advancing toward a crossing in the country’s western border region that provides the only access for U.S. military operations in northern Syria.

A protracted fight over border crossings could severely disrupt U.S. military activity in neighboring Syria. It also could strain the ability of aid organizations to provide desperately needed supplies to the nearly 300,000 civilians who fled fighting in the Syrian city of Raqqa, which U.S.-backed forces reclaimed from the Islamic State militant group this month.

The fresh assault in northern Iraq came as Prime Minister Haider al-Abadi dismissed a Kurdish offer for a cease-fire amid tensions over the Kurds’ referendum last month in favor of independence.